Since we have 

$\Gamma(1/2+it)=\sqrt{\pi/\cosh(\pi t)}\exp[i(2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2)))]$

where $\vartheta(t)$ is the Riemann Siegel function. 
The zeros on the critical line have ordinates the zeros of

the cosine or sine of  the real function

$2 \vartheta(t)+t \log(2\pi)+\arctan(\tanh(\pi t/2))$

But there are real zeros, for example there is one at $s = 4.0260426340124070065475\dots$